# Thread: The Exchange Paradox and its interpretations by Ni, Ti, etc.

1. TL;DR version: If one's reaction to the problem is "It is wrong to use expectation values, here," then one tends to find the problem trivial. If one's reaction to the problem is "Why can't one apply an expectation value, here?" then one has a much lengthier deliberation and discussion.

2. When there's a 'paradox' in applying these sorts of mathematical concepts, it often means one of the following:
1. The 'hammer' (e.g. expected value) that we use to nail down the problem can't actually apply--because of some minor technicality or assumption that we're not used to seeing when we approach problems using that hammer. Essentially, we're so focused on seeing nails, but we're actually looking at a screw. Or.. whatever, use an analogy more poignant than that.
2. Related, and sometimes as a result of the above, we translate a word problem into a mathematical structure improperly--our definition of the problem is flippin' wrong.
3. Our logical and/or mathematical structures are incomplete or inconsistent (something that we don't often accept). This is when we must build new, fancier tools, which often means approaching the problem in a new way, shifting our assumptions, etc.

I accepted your first answer as 'complete enough.' However, I did look into the paper, the Wikipedia article, and other resources out of curiosity--but not because I felt that the detail was necessary. However however, my curiosity was driven by the fact that my master's studies focused on value and decision theory, and I find trying to frame decisions to be fascinating and useful. Hell, half of my career has been built around framing decisions, so I've got a vested interest in and curiosity about what's out there.

Ah, Ne

If we want to tie this to type, I would imagine that stereotypical Ti types would find the lynchpin, the Jenga block that stabilizes the whole structure, that knocks the whole argument in the OP's line of reasoning down--and pull it. They would then dust off their hands, leave the scattered mess on the floor and say, "Welp, my work here is done," and then walk away.

3. It seemed obvious to me that it was a matter of luck, even after knowing the content of one envelope.
I liked the 2nd answer better: ''1.5x expectancy''
The fact that I'm a poker fan probably made a difference.

4. I didn't look at the answers, but since this is about probability, isn't expectation value only used when you want to analyze the amount you would have if you grabbed the envelope X amount of times and want to know the average amount you would end up per grab, for a given number of X trials?

dsfjsdalgjsdf I am now Satan. Good day.

5. I don't know. Actually I don't get it. Because a coin flip is always 50-50. If the envelopes are randomized each trial, then there's no way to assume that the average value of the grabs will converge anywhere specific in between A and 2A.

I guess. I don't know. I'm going to look at your answers now.

6. I am most satisfied with the second explanation, and it is the one that automatically occurred to me when I read the scenario description. The first seems imprecise; the third overelaborate.

You lost me the second any kind of math entered into the...er...equation. Sad, I know.
Yeah same, I just read up on it and basically came to something similar to the first solution, but that depends upon whether I understood it properly.

I also agree with @bologna about people's level of understanding being important to what explanation would work best; mine is limited and simple so a simple explanation is best.

So can I ask if I understood this correctly: The idea is to work out which envelope contains which amount of money and then be offered a switch, so then you need to work out whether you should switch based upon 'expectation' of what the other one might contain? I suppose there isn't really an answer, it's just a 50/50 chance, right?

Meh im not very intelligent so you might have to hand hold me through this one.

8. I like the second answer best, as it uses nice simple algebra rather than expecting me to know exactly how an expectation values is defined! I have more reasoning skills than knowledge, you know.

9. Originally Posted by AffirmitiveAnxiety
Yeah same, I just read up on it and basically came to something similar to the first solution, but that depends upon whether I understood it properly.

I also agree with @bologna about people's level of understanding being important to what explanation would work best; mine is limited and simple so a simple explanation is best.

So can I ask if I understood this correctly: The idea is to work out which envelope contains which amount of money and then be offered a switch, so then you need to work out whether you should switch based upon 'expectation' of what the other one might contain? I suppose there isn't really an answer, it's just a 50/50 chance, right?

Meh im not very intelligent so you might have to hand hold me through this one.
The basic idea is that if you were given the situation where the 2nd amount was determined after you see the first amount, then the concept of "expectation value" applies. In simple terms, the expectation value is the "average" of all possible results. The expectation value of a fair roll of a 6-sided die is "3.5", even though there is no 3.5 on the die, for example.

The expectation value is useful when that concept of average winnings actually means something. And it's more than just how likely you are to win. For instance, I could have a 1/3 chance to win and a 2/3 chance to lose, but if I only pay \$3 to play, and get \$12 if I win, then the average amount I'd win per play is 1/3(1x\$12+3x(-3)) = \$1. Don't worry too much about the math, it depends a lot on specific wording. My point is that even though you're more likely to lose, on average, you still win, because the winnings when you win, on average, outstrip what you lose when you don't win.

Expectation value isn't foolproof. It is possible (google the St. Petersburg paradox) to have an infinite expectation value, but you'd never want to play because you'd run out of money before you ever won anything. (An infinite expectation value is why betting tables have a maximum possible bet, since you could just double your bet each time and eventually win, assuming you could borrow and bet any amount of money.) It is also possible, as in this thread's paradox, to simply misapply the math.

In the case of this paradox, you don't know what you're taking the average of in the first place. It's deliberately trying to trick you with fake reasoning by pretending that you're averaging doubling vs halving - the 1/2(A/2 + 2A) = 5/4 A is the average of doubling vs halving the value A - except you don't really know what A is. (A isn't the value in the envelope you see, it's that other number, X, that you see in the 2nd case, and you aren't really "doubling" or "halving" X.)

But yes, I suspect that the math aspect makes it hard to use for typing, because if one isn't comfortable with the math, one will go for the least mathematical explanation, regardless of type.

10. Originally Posted by uumlau
The basic idea is that if you were given the situation where the 2nd amount was determined after you see the first amount, then the concept of "expectation value" applies. In simple terms, the expectation value is the "average" of all possible results. The expectation value of a fair roll of a 6-sided die is "3.5", even though there is no 3.5 on the die, for example.

The expectation value is useful when that concept of average winnings actually means something. And it's more than just how likely you are to win. For instance, I could have a 1/3 chance to win and a 2/3 chance to lose, but if I only pay \$3 to play, and get \$12 if I win, then the average amount I'd win per play is 1/3(1x\$12+3x(-3)) = \$1. Don't worry too much about the math, it depends a lot on specific wording. My point is that even though you're more likely to lose, on average, you still win, because the winnings when you win, on average, outstrip what you lose when you don't win.

Expectation value isn't foolproof. It is possible (google the St. Petersburg paradox) to have an infinite expectation value, but you'd never want to play because you'd run out of money before you ever won anything. (An infinite expectation value is why betting tables have a maximum possible bet, since you could just double your bet each time and eventually win, assuming you could borrow and bet any amount of money.) It is also possible, as in this thread's paradox, to simply misapply the math.

In the case of this paradox, you don't know what you're taking the average of in the first place. It's deliberately trying to trick you with fake reasoning by pretending that you're averaging doubling vs halving - the 1/2(A/2 + 2A) = 5/4 A is the average of doubling vs halving the value A - except you don't really know what A is. (A isn't the value in the envelope you see, it's that other number, X, that you see in the 2nd case, and you aren't really "doubling" or "halving" X.)

But yes, I suspect that the math aspect makes it hard to use for typing, because if one isn't comfortable with the math, one will go for the least mathematical explanation, regardless of type.
Yeah that does make a bit more sense, but as you said, the maths side of things is difficult for me and I cannot understand the maths at all. I suppose I should watch my brother play on a fruit machine and try to work it out from that /badjoke.

You've intrigued me now though, im probably going to spend hours looking over this topic and even trying to understand the maths behind it. I can't help it, if im ignorant of something I have no problem admitting it but I desire to understand it so as to broaden my understanding.

Ive never understood why people cant admit to their igorance on something. I suppose it has something to do with hubris, but to me it always causes more harm than good.

Anyhow thanks for the explanation and your patience.

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